Tried to find a API to access these images but can't seem to find any. Anyone know if there is any API?
Many other institutions have open APIs for their collection data, and often for images as well. See:
Cooper-Hewitt (data released on github) Brooklyn Museum (API) Powerhouse Museum (API) Rijksmuseum Indianapolis Museum of Art (also data on github) Europeana (a EU-wide consortium) DPLA (similar to Europeana, but in the US)
Also, lots of other institutions have open access to their images - many much more comprehensive than The Getty (not to sound bitter, but Getty is getting a lot of press for this where others have lead for years). It's really an imperative at this point - and for most institutions more a matter of when (since few have substantial technical staff) than if. The Getty is an exception, having a large, well-funded staff, but tends to be conservative in policy changes.
For other examples of cultural institutions sharing images widely:
National Gallery (high rez) Los Angeles County Museum of Art The Met
All this just off the top of my head -
For example, the National Gallery has high-resolution scans, but they aren't downloadable (perhaps you can scrape them, but that isn't accessible to most people). LACMA has only low-resolution images.
Off the rest of your list, Rijksmuseum and the Brooklyn Museum have only low-resolution images, Cooper-Hewitt and Europeana have almost nothing (only very low-resolution). I haven't checked everything.
Maybe you are right that it is becoming more common, but I can't see that myself. The Met is the only example I know of that compares. The Met does have decent resolution downloads available, but the Getty's downloadable scans are still much better.